7.3. Singular Kähler-Einstein metrics of negative curvature [02FX]

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7.3. Singular Kähler-Einstein metrics of negative curvature

Theorem 7.8.

Let $V$ be a general type projective algebraic variety with only canonical singularities such that $K_{V}$ is ample. Let $h^{N}$ a smooth hermitian metric on $\omega^{N}_{V}$ such that $\Omega=c_{1}(K_{V},h)$ is a smooth Kähler form on $V$ .

There is a unique $\varphi\in{\mathcal{C}}^{0}(V,\mathbb{R})$ such that:

(1)

$\varphi$ is $\Omega$ -psh.
(2)

$\Omega+dd^{c}\varphi$ semi Kähler current with ${\mathcal{C}}^{0}$ potential.
(3)

$(\Omega+dd^{c}\varphi)^{n}=e^{\varphi}v(h)$ .

Consequently $\Omega+dd^{c}\varphi$ is the unique singular KE metric on $V$ of negative curvature in the canonical class of $V$ . The current $\Omega+dd^{c}\varphi$ has continuous potentials and is smooth on $V^{reg}$ where it defines a bona fide KE metric.

Proof.

This is a consequence of Theorems 4.1, 4.4, and Definition 7.3. ∎

Remark 7.9.

Thanks to Theorem 5.4, for $X$ a projective algebraic manifold of general type such that $R(X):=\oplus_{n\in\mathbb{N}}H^{0}(X,\mathcal{O}_{X}(nK_{X}))$ is finitely generated, $X$ has a unique birational model $V$ such that the above hypotheses hold. Thus we have a birational map $\pi:X\dashrightarrow V$ which is well defined outside an indeterminacy locus $S$ of codimension $\leq 2$ . In particular $\pi^{*}(\omega+dd^{c}\varphi)$ is a closed positive current on $X-S$ that extends to a closed positive current $T$ on $X$ itself. The current $T$ defines a KE metric on $X-S$ . It needs not be a singular KE metric on $X$ though, since its potentials may have logarithmic poles on $S$ , in fact algebraic singularities of the form $\alpha\log(\sum|f_{i}|^{2})+O(1)$ $f_{i}$ holomorphic and $\alpha\in\mathbb{Q}_{>0}$ . Moreover, $T$ lies in the canonical class of $X$ iff $X$ is a smooth minimal model as in [Ts].

Connection with [Ts]

Let $X$ be a complex projective manifold such that $K_{X}$ is nef and big. Let $\Omega$ be a smooth Kähler metric on $X$ and consider the Kähler-Ricci flow

\frac{\partial\Omega_{t}}{\partial t}=-Ric(\Omega_{t})-\Omega_{t},\ \ \Omega_{0}=\Omega.

In [Ts], it was proved that this flow has a global solution for all time $t\in[0,\infty[$ , and an argument was given, recently fully completed in [TZ], to the effect that $\Omega_{t}$ converges to a closed positive current $T_{KE}$ , independent of $\Omega$ , which defines a smooth Kähler-Einstein metric outside the exceptional divisor $E$ of the holomorphic bimeromorphic map $X\to X_{can}$ . Its potential satisfies the Monge Ampère équation considered in Theorem 7.8 outside $E$ . It follows from proposition 4.4 that the current $T_{KE}$ coincides with the solution produced by Theorem 7.8.

The notes [ST], [TZ] announce a proof of the following properties, already conjectured by [Ts], that $T_{KE}$ has locally bounded potential and satisfies the degenerate Monge-Ampère equation considered in Theorem 7.8. Our Theorem 7.8 in this case gives the precision that $T_{KE}$ has continuous potentials.

Example 7.10.

A nodal sextic threefold is of general type, Gorenstein, terminal, is its own canonical model, has no smooth minimal model and does not have quotient singularities. Therefore the orbifold method of [Ko] does not work and [Ts] does not apply.